Relativity at first order and gravitational deflection of light - An interplay between history and didactics
Résumé
Einstein's failure to derive in his Zurich notebook the equations of general relativity in vacuum was, as well documented, mainly due to his 1912 metric which ignored the influence of gravitation on its spatial part, even at first order. We argue that a complete application of the equivalence principle using relativity at first order in V/c would have enabled him to obtain the correct deflection of light (as well as probably Mercury's perihelion advance) as soon as 1911. The key point is that the transformations x'=x-Vt, t'=t-Vx/c² (Lorentz 1895) allow to discuss not only time issues (as Einstein did in 1907 and 1911) but also issues concerning space; in particular, they imply not only a "time dilation" T'=T(1-Vv/c²) for moving clocks but also a "length contraction" l'=l(1+Vv/c²) for moving rods, two inverse relations leading simply to the correct metric at first order. More generally, these transformations present today a great pedagogical interest for the teaching of special relativity at an elementary level.
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